Closed form micro-macro relationships for periodic masonry

Abstract

The use of theoretical models for providing a constitutive identification of masonry, starting from the individual properties of the phases (i.e. mortar and bricks), constitutes an attractive alternative to costly experimental investigations. In the case of brickwork with periodic texture, the latter issue is tackled resorting to the homogenization theory by Anthoine [1] and solved by means of the finite element method. In order to obtain closed form formulations many authors in literature made simplifying assumptions regarding either the masonry bond, Pande et al. [2], or the joints thickness, Cecchi & Sab [3]. The simplifications adopted turn out to introduce significant errors in the results when either a large difference in stiffness between the phases or a non negligible thickness of the joint are encountered. In the present paper a homogenization procedure is presented, which takes into account the effect of the bond and the Poisson-type interaction between mortar and brick. Assuming a simplified kinematics for the phases belonging to the R.V.E., the so-called localization problem is solved by imposing the minimization of the average internal strain energy. Closed form formulations are then derived for the equivalent in-plane elastic constants of masonry. The expressions found are consistent with those obtained in literature in the limit cases in which masonry is tackled as a stratified medium or where the joints are treated as interfaces. The accuracy of the results is investigated by means of a comparison with finite element analysis. A parametric study, conducted varying the geometries and the mechanical properties of the phases, shows that the error introduced over a very wide range of values for the elastic properties is lower than 8%, meaning that the procedure is ready to be used for non-linear analysis.